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A066760 a(n) = Sum_{1<=k<=n, k is not a divisor of n and k is not coprime to n} k.

%I A066760 #41 Jan 06 2024 23:35:38
%S A066760 0,0,0,0,0,4,0,6,6,18,0,27,0,40,37,42,0,79,0,89,74,108,0,145,45,154,
%T A066760 96,183,0,274,0,210,184,270,163,360,0,340,257,411,0,556,0,467,418,504,
%U A066760 0,669,140,683,439,657,0,880,369,805,548,810,0,1183,0,928,779,930,502
%N A066760 a(n) = Sum_{1<=k<=n, k is not a divisor of n and k is not coprime to n} k.
%C A066760 This function can be used to prove no p^k is perfect or multi-perfect.
%H A066760 Michael De Vlieger, <a href="/A066760/b066760.txt">Table of n, a(n) for n = 1..10000</a>
%F A066760 a(n) = 1 + n*(n+1)/2 - sigma(n) - n*phi(n)/2.
%F A066760 a(n) = 0 if and only if n = 1, 4 or a prime. - _Robert G. Wilson v_, Jul 31 2004
%F A066760 a(n) = 1 + A067392(n) - A000203(n). - _Amiram Eldar_, Dec 03 2023
%F A066760 a(n) = Sum_{k=1..n} k * (1 - floor(1/gcd(n,k))) * (ceiling(n/k) - floor(n/k)). - _Wesley Ivan Hurt_, Jan 06 2024
%e A066760 There are three integers that satisfy this definition for n = 12, namely 8, 9 and 10. These sum to 27, hence a(12) = 27.
%p A066760 f:= n -> 1 + n*(n+1)/2 - numtheory:-sigma(n) - n*numtheory:-phi(n)/2;
%p A066760 0, seq(f(n),n=2..100); # _Robert Israel_, Nov 02 2014
%t A066760 Table[n(n + 1)/2 + 1 - EulerPhi[n] * n/2 - DivisorSigma[1, n], {n, 2, 65}] (* _Robert G. Wilson v_, Jul 31 2004 *)
%t A066760 Table[Sum[k * Boole[Not[Divisible[n, k]]] * Boole[GCD[n, k] > 1], {k, n - 1}], {n, 65}] (* _Alonso del Arte_, Nov 02 2014 *)
%o A066760 (PARI) a(n)=n*(n + 1 - eulerphi(n))\2 + 1 - sigma(n) \\ _Charles R Greathouse IV_, Nov 02 2014
%o A066760 (Magma) [0] cat [1 + n*(n+1)/2 - SumOfDivisors(n) - n*EulerPhi(n)/2: n in [2..70]]; // _Vincenzo Librandi_, Nov 03 2014
%Y A066760 Cf. A000010, A000203, A000217, A023896, A024816, A045763, A067392.
%K A066760 nonn,easy
%O A066760 1,6
%A A066760 _Jon Perry_, Jan 17 2002
%E A066760 Offset corrected to 1 by _Michael De Vlieger_, Jul 05 2014